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RICAM Special Semester on Optimization
Workshop 2
Optimal control and optimization for nonlocal models
Book of Abstracts
(c) Johann Steininger
Linz, AustriaOctober 28–30, 2019
LECTURE I: Introduction to fractional calculusAbner J. Salgado
Univeristy of Tennessee, TN, [email protected]
Abstract
We will discuss several approaches to extend the notion of a derivative to a fractionalorder, their meaning, limitations, and possible applications. For some of these, we will studythe existence, uniqueness, and regularity of solutions to initial boundary value problems withthese operators, and their implications for optimization and numerical approximations.
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LECTURE II: Introduction to nonlocal operators and equationsTadele Mengesha
The University of Tennessee, [email protected]
Abstract
Nonlocal models are becoming commonplace across applications. Typically these modelsare formulated using integral operators and integral equations, in lieu of the commonly useddifferential operators and differential equations. In this introductory lecture, I will presentmain properties of these nonlocal operators and equations. Tools of analyzing them suchas nonlocal calculus, methods of showing well posedness of models, as well as correspondingsolution/function spaces will be discussed. Behavior of solutions of nonlocal equations asa function of a measure of nonlocality will be analyzed. For example, with proper scaling,we will confirm consistency of models by showing that in the event of vanishing nonlocality,a limit of such solutions solve a differential equation. We will demonstrate all these in twomodel examples: a nonlocal model of heat conduction and a system of coupled equations fromnonlocal mechanics.
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On the properties of the nonlocal Cahn–Hilliard model with the double obstacle potentialOlena Burkovska
Florida State University, FL, USA,[email protected]
Abstract
We consider a nonlocal Cahn–Hilliard model with the double-well obstacle potential on abounded domain, which is motivated by the problem of phase separation in copolymer melts.While different types of nonlocality can appear in the model, we study the case that includesboth local and nonlocal operators. The latter one corresponds to a nonlocal Ginzburg-Landauenergy functional and describes long-range interactions between particles in the model. Wealso discuss several ways how to prescribe nonlocal boundary fluxes that are analogous to thelocal Neumann boundary conditions.
We analyze structural properties of the solution and derive suitable discretization tech-niques. In contrast to the local Cahn–Hilliard problem, which always leads to a diffuseinterface, the proposed nonlocal model can lead to a strict separation into pure phases of thesubstance for nontrivial interactions. Concretely, we show that a sharp interface can alreadyoccur for certain critical (non-zero) value of the interface parameter. Here, the choice of theobstacle potential plays an important role in our analysis, since it guarantees the strict sep-aration into pure phases. However, the corresponding additional inequality constraints arechallenging for the numerical discretization. Mathematically, they lead to a coupled-systeminvolving nonstandard parabolic variational inequalities. We employ implicit Euler time step-ping scheme combined with a finite element discretization in space. The computation of thesolution at each time instance can be restated as a constrained optimization problem, whichwe solve by using the primal-dual active set strategy.
Finally, we provide various one and two dimensional numerical experiments to verify ourtheoretical results and to conduct a comparative study of the nonlocal problem and its locallimit, corresponding to a vanishing radius of nonlocal interactions.
This work is in collaborative work with Max Gunzburger (Florida State University, FL,USA).
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nPINNs: nonlocal Physics-Informed Neural NetworksMarta D’Elia
Sandia National Laboratories, NM, [email protected]
Abstract
Nonlocal models provide an improved predictive capability thanks to their ability to cap-ture effects that classical partial differential equations fail to capture. Among these effects wehave multiscale behavior (e.g. in fracture mechanics) and anomalous behavior such as super-and sub-diffusion. These models have become incredibly popular for a broad range of appli-cations, including mechanics, subsurface flow, turbulence, plasma dynamics, heat conductionand image processing. However, their improved accuracy comes at a price of many modelingand numerical challenges.
In this work we focus on the estimation of model parameters, often unknown, or subjectto noise. In particular, we address the problem of model identification in presence of sparseand possibly noisy measurements. Our approach to this inverse problem is based on thecombination of 1. Machine Learning and Physical Principles and 2. a Unified NonlocalVector Calculus and Versatile Surrogates such as neural networks (NN). The outcome is aflexible tool that allows us to learn existing and new nonlocal operators.
We refer to our technique as nPINNs (nonlocal Physics-Informed Neural Networks); here,we model the nonlocal solution with a NN and we solve an optimization problem where weminimize the residual of the nonlocal equation and the misfit with measured data. The resultof the optimization are the weights and biases of the NN and the set of unknown modelparameters.
In this talk we briefly present the unified vector calculus, introduce nPINNs, describe itsproperties, and provide several numerical results that illustrate our findings.
This is joint work with George E. Karniadakis and Guofei Pang (Brown University) andMichael Parks (Sandia National Laboratories, NM).
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Optimal parameter selection in nonlocal image denoisingJuan Carlos De los Reyes
MODEMAT, Escuela Politecnica Nacional, Quito, Ecuador,[email protected]
Abstract
We propose a bilevel learning approach for the determination of optimal weights in non-local image denoising. We consider both spatial weights in front of the nonlocal regularizer,as well as weights within the kernel of the nonlocal operator. In both cases we investigate thedifferentiability of the solution operator in function spaces and derive a first order optimalitysystem that characterizes local minima. For the numerical solution of the problems, we pro-pose a second- order optimization algorithm in combination with a finite element discretizationof the nonlocal denoising models.
This is joint work with Andres R Miniguano Trujillo (Escuela Politecnica Nacional, Quito,Ecuador) and Marta D’Elia (Sandia National Laboratories, NM, USA).
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Machine Learning of Space-Fractional Differential EquationsMamikon Gulian
Sandia National Laboratories, NM, USA,[email protected]
Abstract
Data-driven discovery of differential equations has recently been approached by embeddingthe discovery problem into a Gaussian Process regression of spatial data, treating unknownequation parameters as hyperparameters of a modified physics-informed Gaussian Processkernel. This kernel includes the parametrized differential operators applied to a prior covari-ance kernel. We extend this framework to linear space-fractional differential equations. Ourmethodology is compatible with a variety of fractional operators and stationary covariancekernels, including the Matern class. The fractional physics-informed GP kernel is given byd-dimensional Fourier integral formulas amenable to generalized Gauss-Laguerre quadrature.
The method allows for discovering fractional-order PDEs for systems characterized byheavy tails or anomalous diffusion, which are of increasing prevalence in physical and financialdomains. A single fractional-order archetype allows for a derivative of arbitrary order to belearned, with the order itself being a parameter in the regression. Thus, the user is not requiredto assume a ”dictionary” of derivatives of various orders, and directly controls the parsimonyof the models being discovered. We illustrate on several examples, including fractional-orderinterpolation of advection-diffusion and modeling relative stock performance in the S&P 500with alpha-stable motion via a fractional diffusion equation.
Joint work with Maziar Raissi (Nvidia), Paris Perdikaris (University of Pennsylvania), andGeorge Karniadakis (Brown University).
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Control and parameter identification for nonlocal models of diffusion and mechanicsMax Gunzburger
Florida State University, FL, [email protected]
Abstract
Optimal control and parameter identification problems constrained by nonlocal steadydiffusion equations that arise in several applications are studied. The control is the right-hand side forcing function, the parameter to be identified is a coefficient function, and theobjective to be minimized is a standard matching functional. A calculus for nonlocal operatorsis used to demonstrate the existence and uniqueness of solutions of the problems and toderive optimality systems. We also demonstrate the convergence, as the extent of nonlocalinteractions vanishes, to corresponding PDE problems. Error estimates for continuous anddiscontinuous finite element discretizations are derived. Numerical examples are providedthat show that nonlocal models provide, compared to local PDE models, superior matchingto target functions with a lower cost of control and also better estimates for non-smooth anddiscontinuous coefficients.
This joint work with M. D’Elia (Sandia National Laboratories, NM, USA).
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Learning nonlocal regularization operatorsGernot Holler
University of Graz, Graz, Austria,[email protected]
Abstract
We aim at learning an operator from a class of nonlocal operators that is optimal for theregularization of an ill-posed inverse problem. The considered class of nonlocal operators ismotivated by the use of fractional order Sobolev semi-norms as regularization operators. Firstfundamental results from the theory of regularization with local operators are extended to thenonlocal case. Then a framework based on a bilevel optimization strategy is developed whichallows to choose nonlocal regularization operators from a given class which i) are optimalwith respect to a suitable performance measure on a training set, and ii) enjoy particularlyfavorable properties. Finally, numerical experiments are presented.
The talk is based on joint work with Karl Kunisch, University of Graz, Austria.
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Some inverse problems for subdiffusion equationsBarbara Kaltenbacher
Alpen-Adria-Universitat Klagenfurt, Austria,[email protected]
Abstract
Reaction-diffusion equations are among the most common partial differential equationmodels. They arise as the combination of two physical processes: a driving force f(u) thatdepends on the state variable u and a diffusive mechanism that spreads this effect over aspatial domain. Application areas include chemical processes, heat flow models and populationdynamics. The direct or forward problem for such equations is now very well-developedand understood, especially when the diffusive mechanism is governed by Brownian motionresulting in an equation of parabolic type. Recently, various anomalous processes have beenused to generalize the case of classical Brownian motion diffusion. Amongst the most popularis one that replaces the usual time derivative by a subdiffusion process based on a fractionalderivative of order α ≤ 1. This leads to a PDE with a nonlocal in time differential operator,with several crucial implications for the related inverse problems of backwards diffusion orcoefficient identification. In this talk, we will present some of these, along with numericalreconstruction methods based on fixed point and Newton type methods.
This is joint work with William Rundell (Texas A&M University, College Station, USA).
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Unique determination of several coefficients in a fractional diffusion(-wave) equation by a singlemeasurementYikan Liu
Hokkaido University, Hokkaido, Japan,[email protected]
Abstract
In this talk, we investigate some inverse problems on determining several coefficients simul-taneously in a nonlocal diffusion process described by a time-fractional diffusion equation bya single measurement on the boundary. With a suitably chosen Dirichlet boundary condition,we prove the uniqueness for these coefficient inverse problems by employing Laplace transform.Further, we generalize some of the above results to the case of Riemannian manifolds.
This is a joint work with Kian Yavar (Aix-Marseille University), Zhiyuan Li (ShandongUniversity of Technology) and Masahiro Yamamoto (The University of Tokyo).
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Time Optimal Control of Linear Fractional SystemsIvan Matychyn
University of Warmia and Mazury in Olsztyn, Poland,[email protected]
Abstract
The presentation provides an overview of the recent results on control of linear fractionalsystems, i.e. systems described by linear fractional differential equations with both constantand variable coefficients. Fractional differential equations involving Riemann–Liouville as wellas Caputo derivatives are under consideration.
The presented results are based on explicit, closed-form solutions of the linear fractionalsystems. Those solutions are expressed in terms of matrix Mittag-Leffler functions in case ofconstant coefficients and generalized Peano–Baker series for variable ones.
Due to importance of the matrix Mittag-Leffler function in the theory of linear fractionalsystems, the problem of its computation is discussed.
Sufficient and necessary conditions for controllability and observability are derived usingfractional Gramian matrices.
Problem of time-optimal control of linear fractional systems is examined using techniqueof reachable sets and their support functions.
A method to construct a control function that transfers trajectory of the system to astrictly convex terminal set in the shortest time is elaborated. The proposed method usestechnique of set-valued maps and represents a fractional version of Pontryagin’s maximumprinciple.
Theoretical results are supported by examples, in which optimal control of “bang-bang”type is obtained.
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Optimal design problems governed by the nonlocal p-Laplacian equationJulio Munoz
Universidad de Castilla-La Mancha, Toledo, Spain,[email protected]
Abstract
It is well-known that nonlocal integral models are suitable to approximate integral func-tionals or partial differential equations. In this talk, a nonlocal optimal design model isgoing to be considered as approximation of the corresponding classical or local optimal designproblem. The new model is driven by the nonlocal p-Laplacian equation, the design is thediffusion coefficient and the cost functional belongs to a broad class of nonlocal functionalintegrals. The purpose is to prove existence of optimal design for the new model. This work iscomplemented by showing that the limit of the nonlocal p-Laplacian state equation convergestowards the corresponding local problem. Also, as in the linear case, the G-convergence of thenonlocal optimal design problem toward the local version is studied. This task is successfullyperformed in two different cases: when the cost to minimize is the compliance functional, andwhen an additional nonlocal constraint on the design is assumed.
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A nonlocal variational model related to the spatially variable fractional LaplacianCarlos N. Rautenberg
George Mason University, VA, US,[email protected]
Abstract
A variational model in weighted Sobolev spaces with non-standard weights is proposedtogether with applications to image processing. The associated weights are, in general, not ofMuckenhoupt type and hence non-standard analysis tools are required. For special cases, theresulting variational problem is known to be equivalent to the fractional Poisson problem. Thediscretized problem is solved via a finite element scheme and for image denoising we proposean algorithm to identify the unknown weights. The approach is illustrated on several testproblems and it yields better results when compared to the existing total variation techniques.
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Sparse optimal control for fractional diffusionAbner J. Salgado
Univeristy of Tennessee, TN, [email protected]
Abstract
We consider an optimal control problem that entails the minimization of a nondifferentiablecost functional, fractional diffusion as state equation and constraints on the control variable.We provide existence, uniqueness and regularity results together with first-order optimalityconditions. In order to propose a solution technique, we realize fractional diffusion as theDirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an equivalentoptimal control problem with a nonuniformly elliptic equation as state equation. The rapiddecay of the solution to this problem suggests a truncation that is suitable for numericalapproximation. We propose a fully discrete scheme: piecewise constant functions for thecontrol variable and first-degree tensor product finite elements for the state variable. Wederive a priori error estimates for the control and state variables.
This is joint work with Enrique Otarola (UTFSM, Chile).
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External optimal control of fractional parabolic PDEsDeepanshu Verma
George Mason University, VA, USA,[email protected]
Abstract
In this talk, we introduce a new notion of optimal control, or source identification ininverse, problems with fractional parabolic PDEs as constraints. This new notion allows asource/control placement outside the domain where the PDE is fulfilled. We tackle the Dirich-let, the Neumann and the Robin cases. The need for these novel optimal control conceptsstems from the fact that the classical PDE models only allow placing the source/control eitheron the boundary or in the interior where the PDE is satisfied. However, the nonlocal behaviorof the fractional operator now allows placing the control in the exterior. We introduce thenotions of weak and very-weak solutions to the parabolic Dirichlet problem. We present anapproach on how to approximate the parabolic Dirichlet solutions by the parabolic Robinsolutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimalcontrol problems has been discussed. The numerical examples confirm our theoretical findingsand further illustrate the potential benefits of nonlocal models over the local ones.
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Shape optimization for identifying interfaces in nonlocal modelsChristian Vollmann
Trier University, Trier, Germany,[email protected]
Abstract
Shape optimization methods have been proven useful for identifying interfaces in modelsgoverned by partial differential equations. For instance, shape calculus can be exploited forparameter identification in models, where the diffusivity is structured by piecewise constantpatches.
On the other hand, nonlocal models, which are governed by integral operators instead ofdifferential operators, have attracted increased attention in recent years. This is due to thelarge variety of applications including, e.g., peridynamics, image processing, nonlocal heatconduction or anomalous diffusion.
In this talk we bring together these two fields by considering a shape optimization problemwhich is constrained by a nonlocal convection-diffusion model. We discuss the constraintequation, derive a novel shape derivative of the associated nonlocal bilinear form and presentnumerical results of the implemented optimization algorithm.
This has been joint work with Prof. Dr. Volker Schulz (Trier University, Trier, Germany).
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A variable-order fractional differential equation and its optimal controlHong Wang
University of South Carolina, Columbia, SC, USA,[email protected]
Abstract
Fractional differential equations were shown to provide competitive modeling capabilitiesof challenging phenomena such as anomalously diffusive transport. Moreover, laboratoryexperiments and field tests showed that the order of fractional differential equations mayvary, e.g., as the structure of the media changes.
We report some preliminary results on the analysis and numerical approximations ofvariable-order fractional differential equations and their optimal control problems.
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Error analysis of fully-discrete finite element approximations for stochastic time-fractional PDEsdriven by space-time white noise
Jilu WangBeijing Computational Science Research Center, Beijing, China,
Abstract
In this work, we consider the convergence rates of numerical methods for solving a stochas-tic time-fractional PDE in a convex polygon/polyhedron. For this model, both the time-fractional derivative and the stochastic process result in low regularity of the solution. Hence,the numerical approximation of such problems and the corresponding numerical analysis arevery challenging. In our work, the stochastic time-fractional equation is discretized by abackward Euler convolution quadrature in time with piecewise continuous linear finite ele-ment method in space for which a sharp-order convergence is established in multidimensionalspatial domains with nonsmooth initial data. Numerical results are presented to illustratethe theoretical analysis. Results on optimal control will also be presented.
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A nonlocal feature-driven exemplar-based approach for image inpaintingClayton Webster
Oak Ridge National Laboratory and The University of Tennessee, Knoxville,[email protected]
Abstract
We will present a nonlocal variational image completion technique which admits simulta-neous inpainting of multiple structures and textures in a unified framework. The recovery ofgeometric structures is achieved by using general convolution operators as a measure of behav-ior within an image. These are combined with a nonlocal exemplar-based approach to exploitthe self- similarity of an image in the selected feature domains and to ensure the inpainting oftextures. We also introduce an anisotropic patch distance metric to allow for better controlof the feature selection within an image and present a nonlocal energy functional based onthis metric. Finally, we derive an optimization algorithm for the proposed variational modeland examine its validity experimentally with various test images.
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Stability and uniqueness in inverse problems for viscoelasticity equations by Carleman estimatesMasahiro Yamamoto
The University of Tokyo, Tokyo, [email protected]
Abstract
The subject is inverse problems of determining spatially varying factors of source termsand coefficients by a finite number of boundary measurement data. Our main result is theconditional stability for such inverse problems and the main tool is Carleman estimates.
We plan to discuss several model equations for the viscoelasticity and one model is thelinear viscoelasticity:
ρ(x)∂2t u(x, t)− Lλ,µu(x, t) +
∫ t
0
Lλ,µu(x, η)dη = F(x, t) in Ω× (−T, T ), (∗)
where u(x, t) = (u1(x, t), u2(x, t), u3(x, t))T , T denotes the transpose of vectors, and λ(x),
µ(x), and λ(x, t, η), µ(x, t, η), and for (a, b) = (λ(x), µ(x)) and (a, b) = (λ(x, t, η), µ(x, t, η)),we define the partial differential operator La,b is defined by
La,bu(x, t) = b∆u + (a+ b)∇divu + (divu)∇a+ (∇u + (∇u)T )∇b.
We use boundary data limited to some suitable sub-boundary and so we need Carleman esti-mate for the Lame system (*) with nonlocal term for functions not having compact supports,which requests a lot of calculus and estimates. On the basis of such Carleman estimates, onecan apply the methodology by Bukhgeim and Klibanov (1981) to prove the stability results.
This talk is based on joint works with Professor O.Y. Imanuvilov (Colorado State Univer-sity), and Professors P. Loreti and D. Sforza (Sapienza University of Rome).
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A probabilistic numerical scheme for nonlocal diffusion equations and its applications to trainingresidual neural networks (ResNets)
Guannan ZhangOak Ridge National Laboratory, TN, USA
Abstract
We will present a new probabilistic numerical scheme for a class of semi-linear nonlocaldiffusion equations with integral kernels in three-dimensional irregular domains with volumeconstraints. The nonlocal Fokker-Planck operator is firstly converted to its adjoint and thenapproximated by the mathematical expectation with respect to the associated the stochasticprocess. The volume constraints are handled by utilizing the first exit time of the underlyingstochastic process. Error analysis will be provided to show first-order in temporal discretiza-tion and p-th order convergence in spatial discretization. In addition, we will show preliminaryresults on applying the developed scheme to training nonlocal ResNets with random dropoutsby exploiting the connection between ResNets training and optimal control of nonlocal back-ward Kolmogorovs equations.
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Pointwise-in-Time Error Estimates for an Optimal Control Problem with Subdiffusion ConstraintZhi Zhou
The Hong Kong Polytechnic University, Hong Kong, China,[email protected]
Abstract
In this talk, we will study numerical analysis for a distributed optimal control problem,with box constraint on the control, governed by a subdiffusion equation which involves a frac-tional derivative of order α ∈ (0, 1) in time. The fully discrete scheme is obtained by applyingthe conforming linear Galerkin finite element method in space, L1 scheme/backward Eulerconvolution quadrature in time, and the control variable by a variational type discretization.With a space mesh size h and time stepsize τ , we establish the following order of conver-gence for the numerical solutions of the optimal control problem: O(τmin(1/2+α−ε,1) + h2) inthe discrete L2(0, T ;L2(Ω)) norm and O(τα−ε + `2hh
2) in the discrete L∞(0, T ;L2(Ω)) norm,with any small ε > 0 and `h = ln(2 + 1/h). The analysis relies essentially on the maximalLp-regularity and its discrete analogue for the subdiffusion problem.
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